5x^2-1=7x

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Solution for 5x^2-1=7x equation:


Simplifying
5x2 + -1 = 7x

Reorder the terms:
-1 + 5x2 = 7x

Solving
-1 + 5x2 = 7x

Solving for variable 'x'.

Reorder the terms:
-1 + -7x + 5x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
-1 + -7x + 5x2 = 0

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
-0.2 + -1.4x + x2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -1.4x + 0.2 + x2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -1.4x + x2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -1.4x + x2 = 0 + 0.2
-1.4x + x2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-1.4x + x2 = 0.2

The x term is -1.4x.  Take half its coefficient (-0.7).
Square it (0.49) and add it to both sides.

Add '0.49' to each side of the equation.
-1.4x + 0.49 + x2 = 0.2 + 0.49

Reorder the terms:
0.49 + -1.4x + x2 = 0.2 + 0.49

Combine like terms: 0.2 + 0.49 = 0.69
0.49 + -1.4x + x2 = 0.69

Factor a perfect square on the left side:
(x + -0.7)(x + -0.7) = 0.69

Calculate the square root of the right side: 0.830662386

Break this problem into two subproblems by setting 
(x + -0.7) equal to 0.830662386 and -0.830662386.

Subproblem 1

x + -0.7 = 0.830662386 Simplifying x + -0.7 = 0.830662386 Reorder the terms: -0.7 + x = 0.830662386 Solving -0.7 + x = 0.830662386 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + x = 0.830662386 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + x = 0.830662386 + 0.7 x = 0.830662386 + 0.7 Combine like terms: 0.830662386 + 0.7 = 1.530662386 x = 1.530662386 Simplifying x = 1.530662386

Subproblem 2

x + -0.7 = -0.830662386 Simplifying x + -0.7 = -0.830662386 Reorder the terms: -0.7 + x = -0.830662386 Solving -0.7 + x = -0.830662386 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + x = -0.830662386 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + x = -0.830662386 + 0.7 x = -0.830662386 + 0.7 Combine like terms: -0.830662386 + 0.7 = -0.130662386 x = -0.130662386 Simplifying x = -0.130662386

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.530662386, -0.130662386}

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